Probability Of Picking Two Coins Summing At Least 30 Cents

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Probability, a cornerstone of mathematics, helps us quantify uncertainty and predict the likelihood of events. When we delve into the realm of probability, we often encounter scenarios involving random selections, such as drawing coins from a piggy bank. In this comprehensive guide, we will tackle a classic probability problem: determining the likelihood of picking two coins whose sum is at least 30 cents. This article aims to provide a detailed, step-by-step solution, making it an invaluable resource for students, educators, and anyone with a keen interest in probability.

Problem Statement

Kevin has a collection of coins in his piggy bank, with an equal number of dimes, nickels, and quarters. He engages in a random coin-picking exercise, where he selects a coin, notes its value, replaces it, and then picks another coin. The core question we aim to answer is: What is the probability that the sum of the values of the two coins picked is at least 30 cents?

Key Concepts

Before diving into the solution, it's crucial to understand the foundational concepts of probability:

  • Probability: The measure of the likelihood that an event will occur. It is quantified as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.
  • Sample Space: The set of all possible outcomes of an experiment. In our case, the sample space includes all pairs of coins that can be picked.
  • Event: A subset of the sample space, representing a specific outcome or set of outcomes. Here, our event of interest is the sum of the two coins being at least 30 cents.
  • Independent Events: Events whose occurrence does not affect the probability of each other. The two coin picks in our problem are independent since the first coin is replaced before the second pick.

Identifying the Coins

Kevin has three types of coins in his piggy bank:

  • Dimes: Worth 10 cents each.
  • Nickels: Worth 5 cents each.
  • Quarters: Worth 25 cents each.

Defining the Sample Space

Since Kevin picks a coin, replaces it, and then picks another coin, we need to consider all possible pairs of coins he can pick. The sample space consists of the following pairs:

  1. (Dime, Dime)
  2. (Dime, Nickel)
  3. (Dime, Quarter)
  4. (Nickel, Dime)
  5. (Nickel, Nickel)
  6. (Nickel, Quarter)
  7. (Quarter, Dime)
  8. (Quarter, Nickel)
  9. (Quarter, Quarter)

There are 9 possible outcomes in the sample space.

Identifying Favorable Outcomes

We are interested in the outcomes where the sum of the two coins is at least 30 cents. Let's calculate the sum for each pair:

  1. (Dime, Dime): 10 + 10 = 20 cents
  2. (Dime, Nickel): 10 + 5 = 15 cents
  3. (Dime, Quarter): 10 + 25 = 35 cents
  4. (Nickel, Dime): 5 + 10 = 15 cents
  5. (Nickel, Nickel): 5 + 5 = 10 cents
  6. (Nickel, Quarter): 5 + 25 = 30 cents
  7. (Quarter, Dime): 25 + 10 = 35 cents
  8. (Quarter, Nickel): 25 + 5 = 30 cents
  9. (Quarter, Quarter): 25 + 25 = 50 cents

The favorable outcomes (sum ≥ 30 cents) are:

  • (Dime, Quarter)
  • (Nickel, Quarter)
  • (Quarter, Dime)
  • (Quarter, Nickel)
  • (Quarter, Quarter)

There are 5 favorable outcomes.

Basic Probability Formula

The probability of an event is calculated as:

Probability = (Number of favorable outcomes) / (Total number of possible outcomes)

Applying the Formula

In our case:

  • Number of favorable outcomes = 5
  • Total number of possible outcomes = 9

So, the probability that the sum of the two coins picked is at least 30 cents is:

Probability = 5 / 9

Expressing the Result

The probability can be expressed as a fraction (5/9) or as a decimal (approximately 0.5556) or as a percentage (approximately 55.56%).

Probability of Each Coin Pick

Since Kevin has an equal number of dimes, nickels, and quarters, the probability of picking any specific coin on the first pick is 1/3. Similarly, since the coin is replaced, the probability of picking any specific coin on the second pick is also 1/3.

Independent Events

The two coin picks are independent events because the outcome of the first pick does not affect the outcome of the second pick. The probability of two independent events occurring is the product of their individual probabilities.

Alternative Approach

Another way to approach this problem is by using a probability table. We can create a table showing all possible outcomes and their probabilities:

Dime (1/3) Nickel (1/3) Quarter (1/3)
Dime (1/3) 20 cents 15 cents 35 cents
Nickel (1/3) 15 cents 10 cents 30 cents
Quarter (1/3) 35 cents 30 cents 50 cents

Each cell in the table represents a possible outcome, and the sum of the coins is displayed. The probability of each outcome is the product of the probabilities of the individual coin picks, which is (1/3) * (1/3) = 1/9.

Favorable Outcomes Revisited

From the table, we can see the favorable outcomes (sum ≥ 30 cents) are:

  • (Dime, Quarter)
  • (Nickel, Quarter)
  • (Quarter, Dime)
  • (Quarter, Nickel)
  • (Quarter, Quarter)

Each of these outcomes has a probability of 1/9. To find the total probability, we add the probabilities of the favorable outcomes:

Total Probability = (1/9) + (1/9) + (1/9) + (1/9) + (1/9) = 5/9

Not Replacing the Coin

If the coin was not replaced after the first pick, the probabilities would change for the second pick. This would significantly alter the solution, making the events dependent rather than independent.

Miscalculating Sums

Double-checking the sums of the coins is essential to avoid errors. A simple mistake in addition can lead to an incorrect final probability.

Ignoring Possible Outcomes

Ensuring that all possible outcomes are considered is crucial for an accurate probability calculation. Missing an outcome will lead to an incorrect denominator in the probability fraction.

Misunderstanding Independence

Failing to recognize that the coin picks are independent events can lead to using incorrect formulas or methods to calculate the probability.

In summary, the probability that the sum of the two coins picked by Kevin is at least 30 cents is 5/9. We arrived at this solution by identifying the sample space, determining the favorable outcomes, and applying the basic probability formula. This problem illustrates the fundamental principles of probability and the importance of carefully considering all possible outcomes. Understanding these concepts is crucial for solving more complex probability problems in various fields, including statistics, finance, and game theory. By following the detailed steps and explanations provided in this guide, you can enhance your understanding of probability and confidently tackle similar problems in the future.

To reinforce your understanding, consider solving similar problems with different coin denominations or probabilities. Practice makes perfect, and the more you apply these concepts, the more proficient you will become in probability calculations.

Probability is a fascinating and practical area of mathematics. Mastering the basics, as demonstrated in this coin-picking problem, can unlock the door to more advanced topics and real-world applications. Keep practicing, and you'll find yourself confidently navigating the world of probability.